Re: How to integrate this by pen and pencil?

On 2005-08-27, kiki <lunaliu3@xxxxxxxxx> wrote:
> Hi all,
>
> how do you integrate this indefinite integral?
>
> Integrate(exp(-2x)*(sec(x))^2)

It is not expressible in terms of elementary functions. If you integrate
by parts, the non-trivial parti becomes int(exp(-2x)*tan(x) dx). The two
transcendental functions here are e(x) = exp(-2x) and tan(x).
Liouville's principle states that if an elementary antiderivative
exists, then it's going to be of the form R(x) + ln Q(x), where R and Q
are rational functions in e(x) and tan(x).

If you pick one of the transcendentals, say e(x), you can express both R
and Q in canonical form with respect to it. The canonical form is
obtained by a partial fraction decomposition of R and a factorization
into irreducibles of Q. R = P + F, and Q = G1*G2*G3..., where P is a
polynomial in e(x), F is a sum of irreducible fractions, and G1, G2, ...
are irreducible polynomials in e(x), with the coefficients being
rational functions of tan(x). This gives a complete parametrization
of possible antiderivatives. Now you just take a derivative and try to
match up coeffients with the integrand in the same canonical form.

Basically, since the integrand is a polynomial in e(x), the
antiderivative A has to be a polynomial in e(x) as well. You can further
narrow it down to being a first degree polynomial in e(x), A = f(x)*e(x)
+ g(x). But then, since A' = tan(x)*e(x), g(x) has to be a constant and
f(x) has to satisfy tan(x) = -2f(x) + f'(x).

Now, you repeat the above procedure replacing e(x) by tan(x) and allow
the coefficients to be only real numbers. Since tan'(x) = 1 + tan(x)^2,
the only way to get something of degree 1 in the expression -2f(x) +
f'(x) is to have term of the form ln(1+tan^2(x)) in f(x). But then f'(x)
will not have any logarithms and the logs cannot be cancelled in the
expression -2f(x) + f'(x), which contradicts its equality to tan(x).
This shows that f(x) cannot be a rational function of tan(x). Which
implies that the antiderivative of exp(-2x)*tan(x) cannot be an
elementary function.

This kind of proof is an application of the transcendental version of
the Risch algorithm for integration of elementary functions. Luckily, it
is programmed into most computer algebra systems, so they can do the
work for you if you need to prove that some function has no elementary
antiderivative.

Igor
.

Relevant Pages

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